The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

Interpolating Blaschke Products and Factorization Theorems

Let M(H) be the maximal ideal space of H the Banach algebraof bounded analytic functions on the open unit disk. Let G bethe set of nontrivial points in M(H). By Hoffman's work, G hasdeep connections with the zero sets of interpolating Blaschkeproducts. It is proved that for a closed -separated subset Eof M(H) with E G, there exists an interpolating Blaschke productwhose zero set contains E. This is a generalization of Lingenberg'stheorem. Let f be a continuous function on M(H). Suppose thatf is analytic on a nontrivial Gleason part P(x), f(x) = 0, andf 0 on P(x). It is proved that there is an interpolating Blaschkeproduct b with zeros {z_n}_n such that b(x) = 0 and f(z_n) = 0for every n. This fact can be used for factorization theoremsin Douglas algebras and in algebras of functions analytic onGleason parts.     

Sommes D'Exponentielles et Entiers Sans Grand Facteur Premier

Let S(x,y) be the set S(x,y)= 1 n x : P(n) y, where P(n) denotesthe largest prime factor of n. We study , where f is a multiplicative function. When f=1and when f=µ, we widen the domain of uniform approximationusing the method of Fouvry and Tenenbaum and making explicitthe contribution of the Siegel zero. Soit S(x,y) l'ensemble S(x,y)= 1 n x : P(n) y, désigne le plus grand facteur premier den. Nous étudions , lorsque f est une fonction multiplicative. Quand f=1 et quand f=µ,nous élargissons le domaine d'approximation uniformeenutilisant la méthode développée par Fouvryet Tenenbaum et en explicitant la contribution du zérode Siegel. 1991 Mathematics Subject Classification: 11N25, 11N99.     

A Unicity Theorem for Entire Functions

Let k be a non-negative integer. Suppose that f and g are nonconstantentire functions and that a and b (b a(^k) are small functionsrelated to f and g such that (a,f) + (a, g) > 1. Iff^(k) –b and g^k – b assume the same zeros with the same multiplicities,then f g unless (f – a^(k))(g^(k) – a^(k)) = (b –a(K))2. The problem is related to C. C. Yang's question. A correspondingresult was proved for the case where a 0, b 1, k 1 and theorder of f and g is finite.     

Anzai Skew Products with Lebesgue Component of Infinite Multiplicity

Let f be a 1-periodic C¹-function whose Fourier coefficientssatisfy the condition _n|n|³|f(n|² < . For every R\Q andm Z\{0}, we consider the Anzai skew product T(x, y) = (x +, y + mx + f(x)) acting on the 2-torus. It is shown that T hasinfinite Lebesgue spectrum on the orthocomplement L²(dx) ofthe space of functions depending only on the first variable.This extends some earlier results of Kushnirenko, Choe, Lemaczyk,Rudolph, and the author. 1991 Mathematics Subject Classification28D05.     

On the Poles of Igusa's Local Zeta Function for Algebraic Sets

Let K be a p-adic field, let Z (s, f), sC, with Re(s) > 0,be the Igusa local zeta function associated to f(x) = (f₁(x),..., f_l(x)) [K (x₁, ..., x_n)]^l, and let be a Schwartz–Bruhatfunction. The aim of this paper is to describe explicitly thepoles of the meromorphic continuation of Z (s, f). Using resolutionof singularities it is possible to express Z (s, f) as a finitesum of p-adic monomial integrals. These monomial integrals arecomputed explicitly by using techniques of toroidal geometry.In this way, an explicit list of the candidates for poles ofZ (s, f) is obtained. 2000 Mathematics Subject Classification11S40, 14M25, 11D79.     

Volterra Convolution Operators with Values in Rearrangement Invariant Spaces

The Volterra convolution operator Vf(x) = ^x₀(x–y)f(y)dy,where (·) is a non-negative non-decreasing integrablekernel on [0, 1], is considered. Under certain conditions onthe kernel , the maximal Banach function space on [0, 1] onwhich the Volterra operator is a continuous linear operatorwith values in a given rearrangement invariant function spaceon [0, 1] is identified in terms of interpolation spaces. Thecompactness of the operator on this space is studied.     

Structural Stability of Constrained Polynomial Systems

The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by ad–bc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.     

On Storer Difference Sets

Let p, q be distinct odd primes, and let a, b be positive integers.In this paper we prove that if S(p^a, q^b) is a Storer differenceset with the parameters = p^aq^b, k = (–1)/4 and =(–5)/16,then we have a = b = 1, and , where , and r is a positiveinteger. 1991 Mathematics Subject Classification 05B10.     

Solving two-point boundary value problems with spline functions

We consider a collocation method for the approximation of thesolution of the nonlinear two-point boundary value problem y'(x)=f(x,y(x)), y(a)=A, y(b)=B, using splines of degree m3. The methodwhich we shall use leads to a system of recurrence relationswhich can be solved by Newton's method. By obtaining asymptotic error bounds we verify a conjectureof Khalifa & Eilbeck, i.e. splines of even degree can giveeven better solutions than splines of odd degree in certaincases.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

On Certain Exponential Sums and the Distribution of Diffie-Hellman Triples

Let g be a primitive root modulo a prime p. It is proved thatthe triples (g^x, g^y, g^xy), x, y = 1, ..., p–1, are uniformlydistributed modulo p in the sense of H. Weyl. This result isbased on the following upper bound for double exponential sums.Let >0 be fixed. Then uniformly for any integers a, b, c with gcd(a, b, c, p) = 1.Incomplete sums are estimated as well. The question is motivated by the assumption, often made in cryptography,that the triples (g^x, g^y, g^xy) cannot be distinguished fromtotally random triples in feasible computation time. The resultsimply that this is in any case true for a constant fractionof the most significant bits, and for a constant fraction ofthe least significant bits.     

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.     

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

On Finite and InfiniteCk-A-Determinacy

Let f: (Rⁿ,0) (R^p,0) be a C map-germ. We define f to be finitely,or -, A-determined, if there exists an integer m such that allgerms g with j^mg(0) = j^mf(0), or if all germs g with the sameinfinite Taylor series as f, respectively, are A-equivalentto f. For any integer k, 0 k < , we can consider A' sC^kcounterpart (consisting of C^k diffeomorphisms) A^(k), and wecan define the notion of finite, or -,A^(k)-determinacy in asimilar manner. Consider the following conditions for a C germf: (a_k) f is -A^(k)-determined, (b_k) f is finitely A^(k)-determined,(t) , (g) there exists a representative f : U R^p defined on some neighbourhood U of 0 in Rⁿ such thatthe multigerm of f is stable at every finite set , and (g') every f' with j f'(0)=j f(0) satisfiescondition (g). We also define a technical condition which willimply condition (g) above. This condition is a collection ofp+1 Lojasiewicz inequalities which express that the multigermof f is stable at any finite set of points outside 0 and onlybecomes unstable at a finite rate when we approach 0. We willdenote this condition by (e). With this notation we prove thefollowing. For any C map germ f:(Rⁿ,0) (R^p,0) the conditions(e), (t), (g') and (a) are equivalent conditions. Moreover,each of these conditions is equivalent to any of (a_k) (p+1 k < , (b_k) (p+1 k < ). 1991 Mathematics Subject Classification:58C27.     

Tangential Boundary Behaviour of Harmonic and Holomorphic Functions

Let K be a kernel on Rⁿ, that is, K is a non-negative, unboundedL¹ function that is radially symmetric and decreasing. We definethe convolution K * F by and note from L^p-capacity theory [11, Theorem 3] that, if F L^p, p > 1, then K * F exists as a finite Lebesgue integraloutside a set A Rⁿ with C_K,p(A) = 0. For a Borel set A, where We define the Poisson kernel for = {(x, y) : x Rⁿ, y > 0} by and set Thus u is the Poisson integral of the potential f = K * F, andwe write u=P_y*(K*F)=P_y*f=P[f]. We are concerned here with the limiting behaviour of such harmonicfunctions at boundary points of , and in particular with the tangential boundary behaviour ofthese functions, outside exceptional sets of capacity zero orHausdorff content zero.     

Behaviour of a non-local equation modelling linear friction welding

A non-local parabolic equation modelling linear friction weldingis studied. The equation applies on the half line and containsa non-linearity of the form . For f(u) = e^u, global existence and convergence to a steadystate are proved. Numerical calculations are also carried outfor this case and for f(u) = (– u)^1/a.